Many atomic nuclei possess a magnetic moment. Nuclear magnetic resonance (NMR) is a phenomenon exhibited by this select group of atomic nuclei (termed "NMR active" nuclei), which results from the interaction of the nuclei with an applied, external magnetic field.
The magnetic properties of a nucleus are conveniently discussed in terms of two quantities: the magnetogyric ratio (denoted by the symbol .gamma.); and the nuclear spin (denoted by the symbol I). When an NMR active nucleus is placed in a magnetic field, its nuclear magnetic energy levels are split into (2I+1) non-degenerate energy levels, and these levels are separated from each other by a characteristic energy that is directly proportional to the strength of the applied magnetic field. This phenomenon is called "Zeeman" splitting and the characteristic energy is equal to .gamma.hH.sub.o /2.pi., where h is Plank's constant and H.sub.o is the strength of the magnetic field. The frequency corresponding to the energy of the Zeeman splitting (.omega..sub.o =.gamma.H.sub.o) is called the "Larmor frequency" or "resonance" frequency. Typical NMR active nuclei include .sup.1 H (protons), .sup.13 C, .sup.19 F, and .sup.31 P nuclei. For these four nuclei, the nuclear spin I=1/2, and, accordingly, each nucleus has two nuclear magnetic energy levels.
When a bulk material sample containing NMR active nuclei is placed within a magnetic field, the nuclear spins distribute themselves amongst the nuclear magnetic energy levels in a known manner in accordance with Boltzmann's statics. This distribution results in a population imbalance between the energy levels and a net nuclear magnetization. It is this net nuclear magnetization that is studied by NMR techniques
At equilibrium, the net nuclear magnetization is aligned with the external magnetic field and is time-independent. A second magnetic field perpendicular to the first magnetic field and rotating at, or near, the Larmor frequency can also be applied to the nuclei and this second field disturbs the equilibrium and induces a coherent motion (a "nutation") of the net nuclear magnetization. Since, at conventional magnetic field strengths, the Larmor frequency of typical NMR active nuclei is in the megahertz range, this second field is called a "radio-frequency field" (RF field). The effect of the RF field is to rotate the spin magnetization about the direction of the applied RF field. The time duration of the applied RF field determines the angle through which the spin magnetization nutates and, by convention, an RF pulse of sufficient length to nutate the nuclear magnetization through an angle of 90.degree. or .pi./2 radians, is called a ".pi./2 pulse".
A .pi./2 pulse applied at a frequency near the resonance frequency will rotate a spin magnetization that was aligned along the external magnetic field direction in equilibrium into a plane perpendicular to the external magnetic field. The component of the net magnetization that is transverse to the external magnetic field then precesses about the external magnetic field at the Larmor frequency. This precession can be detected with a resonant coil located with respect to the sample such that the precessing magnetization induces a voltage across the coil. Frequently, the "transmitter" coil employed to apply the RF field to the sample and cause the spin magnetization to nutate and the "receiver" coil employed to detect the resulting precessing magnetization are one and the same coil. This coil is generally part of an NMR probe.
In addition to precessing at the Larmor frequency, the magnetization induced by the applied RF field changes and reverts to the equilibrium condition over time as determined by two relaxation processes: (1) dephasing within the transverse plane ("spin-spin relaxation") with an associated relaxation time, T.sub.2, and (2) a return to the equilibrium population of the nuclear magnetic energy levels ("spin-lattice relaxation") with an associated relaxation time, T.sub.1.
In order to use the NMR phenomenon to obtain an image of a sample, a magnetic field is applied to the sample, along with a magnetic field gradient which depends on physical position so that the field strength at different sample locations differs. When a field gradient is introduced, as previously mentioned, since the Larmor frequency for a particular nuclear type is proportional to the applied field, the Larmor frequencies of the same nuclear type will vary across the sample and the frequency variance will depend on physical position. By suitably shaping the applied magnetic field and processing the resulting NMR signals for a single nuclear type, a nuclear spin density image of the sample can be developed.
When an external magnetic field is applied to a nuclei in a chemical sample, the nuclear magnetic moments of the nuclei each experience a magnetic field that is reduced from the applied field due to a screening effect from the surrounding electron cloud. This screening results in a slight shift of the Larmor frequency for each nucleus (called the "chemical shift" since the size and symmetry of the shielding is dependent on the chemical composition of the sample).
In addition to the applied external magnetic field, each nucleus is also subject to local magnetic fields such as those generated by other nuclear and electron magnetic moments associated with nuclei and electrons located nearby. Interaction between these magnetic moments are called "couplings", and one important example of such couplings is the "dipolar" coupling. When the couplings are between nuclei of like kind, they are called "homo-nuclear couplings". In solids, the NMR spectra of spin=1/2 nuclei are often dominated by dipolar couplings, and in particular by dipolar couplings with adjacent protons. These interactions affect imaging by broadening the natural resonance linewidth and thereby reducing the image resolution.
In order to reduce the effect of such couplings, a class of experiments employs multiple-pulse coherent averaging to continuously modulate the internal Hamiltonians such that, in an interaction frame, selected Hamiltonians are scaled. A subclass of such experiments is designed to reduce the effects of homo-nuclear dipolar couplings by averaging the dipolar Hamiltonian to zero over a selected time period in this interaction frame. The most widely used group of these latter experiments consists of long trains of RF pulses applied in quadrature. Data is sampled between groups of pulses.
Multiple-pulse coherent averaging requires that the spin Hamiltonian be toggled through a series of predetermined states, the average of which has the desired property that the dipolar interaction appears to vanish. If an additional requirement is satisfied that the final Hamiltonian state of the series is equivalent to the first Hamiltonian state of the series, the process can be repeated and the temporal response of the sample can be mapped out successively, point-by-point.
The effects of undesirable interactions can be further reduced by a known technique called "second averaging". Consider an undesirable interaction which has an average direction, .OMEGA..sub.u, and a magnitude, .omega..sub.u, in the toggling frame of the RF pulses. The average Hamiltonian for this undesired interaction is then: EQU H.sub.u =.omega..sub.u .OMEGA..sub.u ( 1)
The rotation average Hamiltonian may similarly be written as: EQU H.sub.r =.omega..sub.r .OMEGA..sub.r ( 2)
Though the principles of second averaging may be applied more generally, here only the case where .OMEGA..sub.r .perp..OMEGA..sub.u and .omega..sub.r &gt;&gt;.omega..sub.u is considered.
The spin system evolves or rotates about the sum of the two average Hamiltonians, H.sub.u and H.sub.r. Since H.sub.r is the larger of the two, the analysis can be simplified by transforming the system to a frame which rotates with H.sub.r and examining the averaging of H.sub.u in this second frame (this double transform is the reason for the name "second averaging"). It is, of course, necessary to transform back to the conventional toggling frame to discover the results of this experiment.
When this latter transformation is made, the averaging follows along the same lines as transforming into the toggling frame, with the second averaged Hamiltonian, H.sub.u, being given by ##EQU1## For the case of interest, the Hamiltonian in the frame of second averaging is ##EQU2## and the second averaged Hamiltonian is, ##EQU3## which vanishes over time. Consequently, the undesired interaction is averaged to zero.
Since the Hamiltonian in the second averaging frame is zero, when it is transformed back to the original observation frame (or toggling frame) the averaged Hamiltonian will acquire a rotational frequency of .omega..sub.r (i.e. the second-averaging frequency). Consequently, the second averaging interaction averages those interaction which are orthogonal to it and at relatively low frequency.
Although second averaging can be used to time average an undesired interaction to zero, there are two problems associated with the technique. First, the line-narrowing efficiency of the technique depends on the sum of the second averaging frequency and the frequency of the interaction of interest (the two of which may be one and the same frequency). The second problem is that the observed resonance frequency is offset by the second averaging frequency. It is also necessary that the interaction of interest be aligned with the second averaging rotation axis since the second averaging technique tends to average interactions which are orthogonal.
Second averaging is of particular interest in certain solid-state imaging experiments, for example, pulsed gradient NMR imaging experiments. In these experiments, the interaction of interest is the gradient induced spin evolution. Since this interaction is produced by the presence of an external field (the gradient field), it may be made time-dependent in all reference frames by modulating the magnetic field over time.
In such solid state imaging experiments, a major problem, as discussed above, is dipolar coupling which broadens the resonance line and, therefore, reduces image resolution. In order to reduce dipolar coupling, it is common in solid-state imaging experiments, to utilize any one of a number of well-known multiple-pulse RF pulse sequences, the effect of which is to time average the dipolar coupling to zero and thereby reduce the line width.
When these prior art pulse sequences are used, they tend to interact with the gradient evolution so that the line width, and therefore, the image resolution, becomes dependent on the gradient strength. This dependency introduces a spatial variation in the resolution so that the images of an object have higher resolution in the center than at the edges of the object. This latter effect is well-known in solid-state imaging.
However, it has been found that the gradient interaction may be decoupled from the line-narrowing efficiency of the multiple-pulse RF pulse cycles by using a pulsed gradient and carefully intercalating the gradient pulses between selected subcycles of the RF pulse cycle. This technique is described in detail in my co-pending application entitled "Method for Improving the Resolution of Solid-State NMR Multiple-pulse Imaging Systems", filed on an even date herewith and assigned to the same assignee as the present invention. The disclosure of that application is hereby incorporated by reference.
In particular, the aforementioned application discloses the use of conventional "time-suspension" multi-pulse RF pulse cycles that time average all interactions in an interaction frame in order to perform line narrowing. The gradient pulses are intercalated into the RF pulse sequence in such a manner that the gradient interaction is decoupled from the RF pulse interaction. Depending on the exact relation between the gradient pulses and the RF pulse sequence, the zero order term, the odd terms and the even terms to second order in the Magnus expansion of the dipolar Hamiltonian can be averaged to zero. When this latter technique is used with the time-suspension RF pulse sequences previously mentioned, the result is obtained that only the gradient interaction is observed in the final result.
It has been found that, further problems remain even with the use of such a decoupling technique. More particularly, when the gradient interaction is near zero, there are a number of undesired interactions associated with imperfections in the experimental realization that dominate the spin dynamics resulting in a false linewidth and resonance frequency. This false line width, in turn, decreases overall resolution. Consequently, an attempt has been made to introduce a second averaging interaction along the same axis as the interaction of interest such that the gradient evolution is not averaged to zero, but these other undesired interactions are averaged to zero in the manner discussed above.
Unfortunately, it has been found that the effect of such a course of action is simply to offset the observed resonance by the second averaging frequency such that the error terms caused by the same undesired interactions are now introduced at the point where the true resonance frequency and the second averaging frequency add up to zero. Consequently, the prior art second averaging technique ends up in shifting the point of reduced line-narrowing efficiency from zero, but it does not eliminate the error terms.
Accordingly, it is an object of the present invention to provide a method for operating a solid-state NMR imaging system so that the resulting images have uniform resolution over the entire image.
It is another object of the present invention to provide a method for operating a solid-state NMR imaging system in which line-narrowing can be carried out without introducing an inhomogeneity in the overall spatial resolution.
It is another object of the present invention to provide a method for operating a solid-state NMR imaging system in which the spatial dependency of the average dipolar Hamiltonian which results from line-narrowing is largely eliminated.
It is still another object of the present invention to provide a method for operating a solid-state NMR imaging system in which the spatial resolution is uniform and which method can be used with a variety of conventional line-narrowing RF pulse sequences.
It is a further object of the present invention to provide a method for operating a solid-state NMR imaging system in which second averaging can be carried out at all frequencies.
It is still a further object of the present invention to provide a method for operating a solid-state NMR imaging system in which second averaging can be carried out and in which the data can be sampled in such a manner that the second averaging frequency is not observed.